I will enumerate various divisorial (cohomological) and geometrical conditions on the endomorphisms and discuss how they affect each other. In particular, I will talk about the cases of Hyperkähler and abelian varieties.
Links:
[1] https://www.mpim-bonn.mpg.de/de/taxonomy/term/39
[2] https://www.mpim-bonn.mpg.de/de/node/3444
[3] https://www.mpim-bonn.mpg.de/de/node/5285